$C^\infty$ Local solutionsof elliptical $2-$Hessian equation in $\mathbb{R}^3$

Guji Tian; Qi  Wang; Chao-Jiang Xu

doi:10.3934/dcds.2016.36.1023

Article Contents

2016, Volume 36, Issue 2: 1023-1039. Doi: 10.3934/dcds.2016.36.1023

This issue Previous Article Existence and stability of a two-parameter family of solitary waves for a 2-coupled nonlinear Schrödinger system Next Article Boundary behavior and asymptotic behavior of solutions to a class of parabolic equations with boundary degeneracy

$C^\infty$ Local solutions of elliptical $2-$Hessian equation in $\mathbb{R}^3$

1.
Wuhan Institute of Physics and Mathematics,Chinese Academy of Sciences, Wuhan 430071
2.
School of Mathematics and Statistics, Wuhan University, Wuhan, 430072
3.
School of Mathematics, Wuhan University, 430072, Wuhan

Received: May 31, 2014

Revised: February 28, 2015

Published: May 2017

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Abstract

In this work, we study the existence of $C^{\infty}$ local solutions to $2$-Hessian equation in $\mathbb{R}^{3}$. We consider the case that the right hand side function $f$ possibly vanishes, changes the sign, is positively or negatively defined. We also give the convexities of solutions which are related with the annulation or the sign of right-hand side function $f$. The associated linearized operator are uniformly elliptic.

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Mathematics Subject Classification: Primary: 35J60; Secondary: 35J70.

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